A phrase introduced by Nelder and Wedderburn in 1972 to describe any model that relates μ, the expected value of the response variable (see regression) Y, to a linear combination of the explanatory variables x1, x2,…, xp using the model
where β1, β2,…, βp are unknown parameters and g is the link function. Examples of link functions are g(μ) = μ (the identity link), and g(μ) = ln(μ) (the logarithmic link). These correspond to random variables having a normal distribution and a Poisson distribution, respectively.
For different families of models that deal with variation in the parameter p of a binomial distribution:
g(p) = log(p)−log(1−p) | (the logistic link, also called the logit link) |
g(p) = log{−log(1−p)} | (the complementary log-log link) |
g(p) = −log{−log(p)} | (the negative log-log link) |
g(p) = tan{π(p−½)} | (the cauchit link, also called the inverse Cauchy link) |
g(p) = Φ−1(p) | (the probit link), |
where Φ denotes the cumulative distribution function of the standard normal distribution and here the logarithms are to base e.